Mathematics: Integrals

Integration is a method of adding values on a large scale, where we cannot perform general addition operations. But there are multiple methods of integration, which are used in Mathematics to integrate the functions. Different integration methods are used to find an integral of some function, which makes it easier to evaluate the original integral. Let us discuss the different methods of integration such as integration by parts, integration by substitution, and integration by partial fractions in detail.

1. Integration by Substitution
2. Integration by Parts
3. Integration Using Trigonometric Identities
4. Integration of Some particular function
5. Integration by Partial Fraction

Integration By Substitution

Sometimes, it is challenging to find the integration of a function; thus, we can see the integration by introducing a new independent variable. This method is called Integration by Substitution.

The given form of integral function (say ∫f(x)) can be transformed into another by changing the independent variable x to t.

Substituting x=g(t) in the function ∫f(x) we get:

dx/dt=g′(t)

or dx = g'(t)

Thus,

I=∫f(x) dx=f(g(t))⋅g′(t)dt

Problem:

Evaluate the integral:

∫2xe^x2dx

Step 1: Identify a substitution

We notice that the exponent x² is the inner function, and its derivative 2x is also present in the integrand. This suggests a substitution:

Let: u=x²

Then:                                                

Step 2: Rewrite the integral in terms of u

Substitute u=x² and du=2xdx into the original integral:

Step 3: Integrate

Now the integral is much simpler:

Step 4: Substitute back

Problem:

Evaluate the integral:

Step 1: Identify a substitution

Notice that the denominator is , and the numerator x is related to the derivative of

Let’s set:

Step 2: Rewrite the integral in terms of u

Substitute:  and

The x in the numerator and denominator cancel:

Step 3: Integrate

Recall that  for

Step 4: Substitute back

Problem:

Evaluate the integral:

Step 1: Choose a substitution

The inner function here is 3x3x3x, which is inside the sine function. We can simplify the integration by substituting:

Let:                        

Step 2: Rewrite the integral in terms of u

Step 3: Integrate

We now integrate sin(u):

Step 4: Substitute back u=3xu = 3xu=3x

Integration By Parts

Integration by parts requires a special technique for the integration of a function, where the integrand function is the product of two or more functions.

Let us consider an integrand function to be f(x)⋅g(x)

Mathematically, integration by parts can be represented as:

Problem:

Evaluate the integral:

Step 1: Identify parts for the formula

Integration by parts is based on the formula:

U=x  then du=dx

Dv=e^x dx then v=e^x

Step 2: Apply the formula

Now plug into the integration by parts formula:

Step 3: Integrate the remaining part

Expression becomes;

Problem:

Evaluate the integral:

Step 1: Choose u and dv

For integration by parts, use:

U=ln(x)                 Dv=xdx                          

Step 2: Apply the formula

Plug these into the integration by parts formula:

Simplify the integrand in the second term:

So the integral becomes:

Step 3: Integrate the remaining part

Step 4: Combine everything

Integration Using Trigonometric Identities

In the integration of a function, if the integrand involves any kind of trigonometric function, trigonometric identities can be used to simplify the function for easier integration. A few trigonometric identities are:

Evaluate the integral:

.

Step 1 :use the trigonometric identity

Step 2: Substitute into the integral :

Final answer is

Evaluate the integral:

Step 1 :use the trigonometric identity

Step 2: Substitute into the integral:

                             final answer is     

Evaluate the integral:

Step 1 :use the trigonometric identity

Step 2: Substitute into the integral:

Final answer is

Integration of some particular functions

Integration of certain functions involves important formulae of integration that can be applied to transform other integrations into the standard form of the integrand. The integration of these standard integrands can be easily found using a direct integration method.

Evaluate the integral of ;

Step 1 we know that the derivative of  is .

Therefore,                                                            Final answer is

Find the integral of

Step 1:

The antiderivative of   is .

Therefore  

Integration by partial fraction

We know that a rational number can be expressed in the form of , where p and q are integers and .Similarly, a rational function is defined as the ratio of two polynomials, which can be expressed in the form of partial fractions:  and

1. Proper partial fractions: When the degree of the numerator is less than the degree of the denominator, then the partial fraction is known as a proper partial fraction.
2. Improper partial fraction: When the degree of the numerator is greater than the degree of the denominator, then the partial fraction is known as an improper partial fraction. Thus, the fraction can be simplified into simpler partial fractions that can be easily integrated.

Example  evaluate the integral.

Step 1: factor the denominator.

Step 2:

Express the function as a sum of simpler functions.

Multiply out terms and solve for A and B:

Comparing the coefficients gives

Step 3: Integrate each term separately.

Example :  evaluate the integrals.

 Factor the denominator.

Multiply out terms and solve for A and B:

 Step 3: Integrate each term separately.